Theorem cfinufil 23813

Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 23778 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
cfinufil	⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4558	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
2		ufilb 23791	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
3	2	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
4		ufilfil 23789	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
5	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (Fil‘𝑋))
6		filfinnfr 23762	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋 ∖ 𝑥) ∈ 𝐹 ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ∩ 𝐹 ≠ ∅)
7	6	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ((𝑋 ∖ 𝑥) ∈ Fin → ∩ 𝐹 ≠ ∅)))
8	7	com23 86	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
9	5, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅)))
10	9	imp 406	. . . . . . . . 9 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
11	3, 10	sylbid 240	. . . . . . . 8 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (¬ 𝑥 ∈ 𝐹 → ∩ 𝐹 ≠ ∅))
12	11	necon4bd 2945	. . . . . . 7 ⊢ (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑋 ∖ 𝑥) ∈ Fin) → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹))
13	12	ex 412	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ Fin → (∩ 𝐹 = ∅ → 𝑥 ∈ 𝐹)))
14	13	com23 86	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
15	1, 14	sylan2 593	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∩ 𝐹 = ∅ → ((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
16	15	ralrimdva 3129	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
17	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18		uffixsn 23810	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ∈ 𝐹)
19		filelss 23737	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
20	17, 18, 19	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → {𝑦} ⊆ 𝑋)
21		dfss4 4220	. . . . . . . . . . 11 ⊢ ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
22	20, 21	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23		snfi 8968	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
24	22, 23	eqeltrdi 2836	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25		difss 4087	. . . . . . . . . . 11 ⊢ (𝑋 ∖ {𝑦}) ⊆ 𝑋
26		filtop 23740	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
27		elpw2g 5272	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
28	17, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
29	25, 28	mpbiri 258	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30		difeq2 4071	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
31	30	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋 ∖ 𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥 ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
33	31, 32	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
34	33	rspcv 3573	. . . . . . . . . 10 ⊢ ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
35	29, 34	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
36	24, 35	mpid 44	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37		ufilb 23791	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
38	20, 37	syldan 591	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
39	18	pm2.24d 151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
40	38, 39	sylbird 260	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
41	36, 40	syld 47	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 ∈ ∩ 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ¬ 𝑦 ∈ ∩ 𝐹))
42	41	impancom 451	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → (𝑦 ∈ ∩ 𝐹 → ¬ 𝑦 ∈ ∩ 𝐹))
43	42	pm2.01d 190	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ¬ 𝑦 ∈ ∩ 𝐹)
44	43	eq0rdv 4358	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)) → ∩ 𝐹 = ∅)
45	44	ex 412	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹) → ∩ 𝐹 = ∅))
46	16, 45	impbid 212	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹)))
47		rabss 4023	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋 ∖ 𝑥) ∈ Fin → 𝑥 ∈ 𝐹))
48	46, 47	bitr4di 289	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∩ cint 4896  ‘cfv 6482  Fincfn 8872  Filcfil 23730  UFilcufil 23784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1o 8388  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fbas 21258  df-fg 21259  df-fil 23731  df-ufil 23786

This theorem is referenced by: (None)